Assessment of the Bill Emerson Memorial Bridge - FTP Directory ...

For a group of piles with identical geometry, the group interaction factor isa summation of those factors from individual piles. Note that the group interactionfactor in horizontal x-direction and y-direction may be different, depending on thenumber and the spacing of piles in each direction.A.3 Group stiffness and damping factorsFigure A.4 shows schematically the plan and cross sections of an arbitrarypile group foundation. This figure will be used to explain and obtain the stiffnessand damping factors of a group of piles in all directions. For vertical vibration, thegroup stiffness (k g ) and damping factors (c g ) can be expressed intozz∑ k ∑ cc gzk gzz = ,z = (α A∑∑α AFor torsional vibration, the group stiffness (k gψ ) and damping factors (c gψ )can be evaluated by= [ k + k ( )]1k g 2 2ψ ψ xx r+ yr∑αA2, c ψ= [cψ+ c x( x r+ y 2∑α Ag 1r)]A.6)(A.7)For translational modes, the group stiffness and damping coefficientsalong x axis (kx g, cx g ) and along y axis (ky g, cy g ) can be determined by∑ k ∑ c ∑ c gxk ∑x = k g yck gxy = c g yx = , , ,y = (A.8)∑α Lx ∑α Lx ∑α LA ∑α LyFor rocking vibration about y axis and about x axis, the stiffness anddamping coefficients (kφ g, cφ g ) and (kθ g, cθ g ) can be respectively evaluated byk k 2 2g zxrk xz c2z ckk = + + − φ x φc c x 2 2g φ z rc xz c2z cc xφ, c+ + −∑α = φ(A.9)φLx∑α Lx2 2k kg θ zyrk yz c2z ckk+ + − 2 2=yθc + c +g θ zyrc xz c−2z ccθ, c∑α = yθ(A.10)θLy∑α LyFor the coupled vibration between translational mode along x axis androtational mode about y axis, the group stiffness and damping coefficients (kxφ g,cxφ g ) can be evaluated by1k g = 1xφ ∑ ( k xφ− k xzc), c gαLxxφ = ∑ ( )α Lxc xφ− c xz c(A.11)Similarly, the group stiffness and damping coefficients for the coupled vibrationbetween translational mode along y axis and rotation al mode about x axis, (kyθ g,cyθ g ), can be expressed into11k g yθ= ∑ ( k yθ− k yzc), c g yθ= ∑ (c yθ− c yzααLyLyc)(A.12)105